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Category Theory (Functional Style) part-1

Category Theory

Category theory is a way to abstract programming, enabling the creation of complex programs that are readable, extensible, and testable. It provides a mathematical abstraction for code.

category consist of type object and morphs

  • morphs interpreted as a function or operation that transforms data from one form to another.

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  • object in functional programming might refer to a data structure that bundles together related information.

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f:a -> b // f is morphs and a is object

morphs might refer to functions or transformations that take one input and produce a modified output without any internal state or side effects these types are known as pure functions.

What is Pure functions

Pure functions are deterministic, meaning they always produce the same output given the same input, and they don't rely on or modify any external state.

what is Composition

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To form Relation between two morphs.

The practice of combining simpler functions to build more complex functions. It involves taking two or more functions and creating a new function by chaining them together, where the output of one function becomes the input of the next function.
for example in js/ts - map,filter,reducer

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f:a->f:b->c
composition - gāˆ˜f
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Functional Style

Rules of Category

1> Composition definition
2> Composition associative
3> Composition Identity

Composition Definition

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If you have two functions composed one after another, it implies that there exists a third arrow starting from the first function and ending at the second function.

f:A -> B and g:B -> C then k = g * f,
where k:A -> C

composition k is defined as mapping elements from A to C by first applying function f to map elements from A to B and then applying function g to map the result from B to C.

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Composition Associative

The order of the Composing function should not matter. Whether you first compose f and g than h or g and h and then f.

f: A -> B, g:B -> C and h:C -> D then,
(f * g) * h === f * (g * h)

it assures us that no matter which order we apply the compositions than final result will be the same.

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Composition Identity

f:A -> A

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It is a function that maps every element of a set
A to itself.

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